Minimality of balls in the small volume regime for a general Gamow type functional

TL;DR

This paper proves that for a class of functionals combining perimeter and a kernel-based interaction, the optimal shape minimizing the functional for small volume is a ball, under certain conditions on the kernel.

Contribution

It establishes the minimality of balls as unique minimizers for a broad class of Gamow-type functionals in the small volume regime.

Findings

Balls are the unique minimizers for the functional when the volume parameter is sufficiently small.

The result holds for kernels that are admissible, radial, and decreasing.

The proof applies to a general class of kernels, extending previous specific cases.

Abstract

We consider functionals given by the sum of the perimeter and the double integral of some kernel $g:\mathbb R^N\times\mathbb R^N\to \mathbb R^+$, multiplied by a "mass parameter" $\varepsilon$. We show that, whenever $g$ is admissible, radial and decreasing, the unique minimizer of this functional among sets of given volume is the ball as soon as $\varepsilon\ll 1$.